Definition:P-adic Valuation/Rational Numbers

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Definition

Let $p \in \N$ be a prime number.


Let the $p$-adic valuation on the integers $\nu_p^\Z$ be extended to $\nu_p^\Q: \Q \to \Z \cup \set {+\infty}$ by:

$\map {\nu_p^\Q} {\dfrac a b} := \map {\nu_p^\Z} a - \map {\nu_p^\Z} b$

This mapping $\nu_p^\Q$ is called the $p$-adic valuation (on $\Q$) and is usually denoted $\nu_p: \Q \to \Z \cup \set {+\infty}$.


Also see

Sources